---
jupyter:
  description: "A Reuleaux triangle turning inside a square drills an almost square\
    \ hole: the union of its positions covers 2\u221A3 + \u03C0/6 - 3, about 98.77\
    \ %, of the square."
---

# Drilling a square hole with a Reuleaux triangle

A Reuleaux triangle has the same width in every direction, so when it turns inside a square, it touches all four
sides at once. A drill bit of that shape, guided to follow it, cuts an almost square hole. Only the corners stay
slightly round, and the hole covers 2√3 + π/6 − 3, about 98.77 %, of the square. The example is a triangle of width
20, built from three arcs and turned through a third of a revolution, and every position is united into the hole.

![A Reuleaux triangle turning inside a square and the almost square hole it drills](../_static/gallery/square-hole.gif){.result}

```python
import math
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
from IPython.display import HTML
import planex as px
```

```python nbsphinx="hidden"
from docs_figures import LIGHT, Row
```

## The triangle

The triangle consists of three arcs, each centered on the opposite corner of an equilateral triangle with the side
`s` = 20 (Figure 1).

```python
s = 20.0
A = (0.0, 0.0)
B = (s, 0.0)
C = (s / 2, s * math.sqrt(3) / 2)
triangle = px.QuickSketch(B).Arc(center=A, sweep=60).Arc(center=B, sweep=60).Arc(center=C, sweep=60).Face()
center = (s / 2, s * math.sqrt(3) / 6)
```

```python nbsphinx="hide-input"
Row((triangle, "Three arcs around the corners"), caption="A Reuleaux triangle of width 20")
```

**Check:** its area is (π − √3) s² / 2:

```python
print("area", triangle.mass, "exact", (math.pi - math.sqrt(3)) / 2 * s**2)
```

## Constant width

Turned to any angle, the triangle's bounding box is s by s. Moving it so that the box sits on the square's lower left
corner places it touching all four sides. **Check:** in 360 positions over a third of a turn, the box keeps its size:

```python
def placed(angle):
    t = triangle.Rotated(center, angle)
    (x0, y0), (x1, y1) = t.bounding_box
    return t.Translated((-x0, -y0)), (x1 - x0, y1 - y0)


boxes = [placed(120 * n / 360)[1] for n in range(360)]
print("bounding box differs from s by at most", max(max(abs(w - s), abs(h - s)) for w, h in boxes))
```

## The hole

The hole is the union of all positions over a third of a turn, which by the triangle's symmetry covers every position
there is. **Check:** with more positions, the scallops between them shrink, and the area approaches the exact value:

```python
exact = (2 * math.sqrt(3) + math.pi / 6 - 3) * s**2
for count in (30, 120, 360, 1440):
    hole = px.Unify([placed(120 * n / count)[0] for n in range(count)])
    print(count, "positions:", round(hole.mass / s**2, 6), "of the square, exact", round(exact / s**2, 6))
```

The hole of 1440 positions in the square, with some of the positions dashed, and one of its round corners:

```python nbsphinx="hide-input"
square = px.QuickSketch().Rectangle(s, s).Face()
positions = [placed(120 * n / 1440)[0].outer for n in range(0, 1440, 160)]
Row((square, "The hole in the square", {"fill": False, "fills": [(hole, LIGHT)], "dashed": positions}),
    (square, "A corner", {"fill": False, "fills": [(hole, LIGHT)], "view": ((-0.2, 3), (-0.2, 3))}),
    caption="The almost square hole")
```

## In motion

The triangle drilling the hole:

```python nbsphinx="hide-input"
fig, ax = plt.subplots(figsize=(5, 5))


def frame(n):
    ax.clear()
    square.Plot(ax=ax, face_col="whitesmoke", legend=False)
    if n:
        drilled = px.Unify([placed(120 * q / 240)[0] for q in range(0, 4 * n + 1, 2)])
        drilled.Plot(ax=ax, face_col="lightsteelblue", legend=False)
    placed(120 * 4 * n / 240)[0].Plot(ax=ax, face_col="tab:orange", legend=False)
    ax.set_aspect("equal", adjustable="box")
    ax.set_xlim(-1, s + 1)
    ax.set_ylim(-1, s + 1)
    ax.axis("off")


animation = FuncAnimation(fig, frame, frames=60, interval=80)
plt.close(fig)
HTML(animation.to_jshtml())
```

```python nbsphinx="hidden"
import sys
sys.path.insert(0, "..")
from gallery import save_scene, save_figure, save_animation, HIDDEN
save_animation(animation, "square-hole")
```